Electromagnets with axially symmetric coil arrangements are commonly used for the generation of uniform magnetic fields. By “uniform field” is meant a magnetic field whose intensity over a desired region does not vary more than 100 parts per million (ppm). An important application nowadays is the medical use of MRI where the desired region is often referred to as the imaging region. MRI of a human body or body part requires large magnets. The magnets may be composed of superconductive materials or of resistive materials. A natural design criterion is that the volume of the coils be as small as possible, consistent with a set of prescribed constraints, since the volume of the coils often correlates closely with a magnet's weight, cost, and power consumption. This is especially important for superconducting magnets since their cost depends strongly on the amount of superconducting wire required, and the necessity for artificial cooling of the magnet coils down to the critical temperature at which superconducting behaviour is achieved.
Many difficulties are encountered in such a magnet design problem, and many solutions have been proposed. The various proposed solutions exhibit one or more disadvantages. For example, the problem of volume minimization of electromagnetic coil systems has been considered previously by Kitamura et al, M. Kitamura, S. Kakukawa, K. Mori, and T. Tominaka, “An optimal design technique for coil configurations in iron-shielded MRI magnets,” IEEE Tran. Magn., vol. 30, no. 4, pp. 2352-2355, 1994, (“Kitamura”), and by Xu et al, H. Xu, S. M. Conolly, G. C. Scott, and A. Macovski, “Homogeneous magnet design using linear programming,” IEEE Trans. Magn., vol. 36, no. 2, pp. 476-483, 2000 (“Xu”). For simplicity, when discussion of a particular reference is needed or desirable, it may be identified by the name of the lead author. The reader is urged if needed to review these references for a more complete discussion of the various known methods and their pros and cons and for a more complete understanding of the methods described herein. While the methods disclosed by Kitamura and Xu have benefits, they also have requirements that are undesirable. For example, the method of Kitamura assumes unidirectional currents in the coils, while the method of Xu requires that the length-to-width ratio of the coils be specified. Therefore, neither approach determines, in general, the minimum volume solution, as is defined in this invention. It is also noted that a variety of procedures for optimizing electromagnets have been described that utilize criteria other than coil volume minimization; for a review of these the reader is to Xu.